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Chapter 8:
Holography and the Wigner Function
There is barely any other invention which had more influence on optical sciences than holography. One of the major steps in developing Gabor’s original concept into a practical method for wavefront recording and reconstruction is marked by the introduction of the offset reference wave by E. Leith and J. Upatnieks. The ability to separate the object wavefront spatially from other parts of the reconstructed wave is the basis for numerous applications of holography. Holography also forms the theoretical framework for many other optical phenomena and techniques. This includes the design of diffractive optical elements as a specialized case of computer-generated holography. Interferometric techniques, which are also based on a coherent superposition of two wavefronts, can be interpreted as a holographic recording of a complex wavefront. Generalized holography, where the wavefront is recorded on arbitrary surfaces, serves as a framework for solving the inverse source problem and for understanding tomographic imaging techniques. The latter is closely related to synthetic aperture radar imaging and acoustic imaging, both of which expand the holographic principle beyond its use with optical radiation. The importance of holography has prompted numerous interpretations focused either on a rigorous analysis or a rather intuitive explanation. In this context, we mention Roger’s explanation of holography, which explains the recorded pattern of a Gabor in-line hologram as the superposition of the plane reference wave and the spherical wave originating from each object point. This essentially interprets the hologram as the superposition of Fresnel zone patterns.Another useful interpretation is the analysis of the Leith–Upatnieks offset hologram as a phase-modulated spatial carrier, where the phase modulation corresponds to the information that is lost in a conventional photographic recording. This interpretation also provides the basis for computer-generated holography, and in particular for the coding of complex amplitudes as detour holograms. This chapter presents an alternative explanation of holography based on the Wigner distribution function (WDF), which was originally published by E.Wigner in 1932 as a phase-space representation of quantum mechanical systems. The French scientist J. Ville would introduce it about 15 years later as a tool for signal processing. It is the fact that WDF combines insight into physical phenomena with signal processing capabilities that makes it unique among other time-frequency transformations. The WDF was introduced into optics primarily by the work of Bastiaans during the late 70s and early 80s.
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